March 12, 2012

… or Agol’s Theorem?

This just in: Ian Agol (UC Berkeley), speaking at the Institut Henri Poincaré today, has announced a proof of the very same Wise’s Conjecture that I blogged about just last week! In particular, this implies the Virtually Haken Conjecture. His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo). It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech).

Of course, to get the statement on the blackboard, you also have to invoke Perelman’s proof of Geometrization.

For the suspicious, I did not know about this when I blogged last week. Thanks to Stefan Friedl (Köln) for the news and the photo.

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The proof of Agol builds, as explained by Henry in his blog entry, on the work of Kahn and Markovic and it also makes use of the results and ideas of Dani Wise.
It follows from Agol’s theorem that given a hyperbolic 3-manifold with fundamental group pi that
(1) pi admits a finite index subgroup which is a quasi-convex subgroup of a right angled Artin group (this follows from Haglund-Wise)
(2) N is virtually fibered (in fact pi is virtually RFRS and the statement of Agol’s virtual fibering theorem holds)
(3) pi is LERF (also known as subgroup separable), this uses work of Haglund and the proof of the tameness conjecture. In fact a stronger statement holds: any geometrically finite subgroup of pi is a virtual retract.
(4) pi is large, i.e. pi admits a finite index subgroup which maps onto a non-cyclic free group, in particular the Betti numbers of finite covers can become arbitrarily large
(5) pi is linear over Z, i.e. pi admits a faithful representation pi to GL(n,Z) for some n
(6) pi is conjugacy separable (this uses work of Minasyan)
(7) pi is virtually biorderable
it seems like every nice property of fundamental groups which one can possibly ask for either holds for pi or a finite index subgroup of pi.

Dave: My understanding from Ian is that he didn’t talk much, if at all, about the argument in the seminar talk pictured above, but that he will do so in his workshop talk at IHP later this month, for which the abstract is in the comments below.

Abstract: We prove that cubulated hyperbolic groups are virtually special. Work of Haglund and Wise on special cube complexes implies that they are therefore linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston’s virtual fibering question. The results depend on a recent result of Wise, the malnormal special quotient theorem; the cubulation of closed hyperbolic 3-manifolds by Bergeron-Wise using the existence of nearly geodesic surfaces by Kahn-Markovic; and a generalization of previous work with Groves and Manning to the case of torsion (which is joint with Groves and Manning).

No, it does not, assuming that by “perfect group” you mean one with trivial abelianization (i.e. every element is a commutator). In fact, there exists many hyperbolic 3-manifolds where the first homology group vanishes, which is equivalent to the fundamental group being perfect. Item (4) does imply that the fundamental group of a hyperbolic 3-manifold has a finite index subgroup which is not perfect, but that was previously known by work of Lubotzky since these are linear groups.

Yes, that is exactly why I was puzzled, since as you yourself pointed out, most hyperbolic 3-manifolds are homology spheres. I did not realize you could have a perfect group with a non-perfect finite index subgroup.

This is amazing progress. What chance these methods would apply to abstract Poincaré duality groups? Is anyone actively working on the PD3-group case? One frustration in the past with PD3-groups has been the lack of an algebraic analogue of Scott’s compact submanifold theorem. Does Scott’s theorem play a crucial role in the Agol–Wise theory?

Peter, the Scott core theorem doesn’t play any role in this as far as I can tell. Instead, the proof crucially relies on the group being Gromov hyperbolic and having a ton of “codimension 1” subgroups (which in the case of hyperbolic 3-manifolds come from Kahn-Markovic). I think Cannon conjectured that a hyperbolic PD3-group is in fact the fundamental group of a hyperbolic 3-manifold, but I guess that’s a separate issue.

There’s some hope that one can perform the Kahn-Markovic construction for hyperbolic PD3 groups to show that they are cubulated. Then virtual specialness should imply that they are 3-manifold groups, by extending the action over the ball in a finite-sheeted cover using the hierarchy. Markovic pointed out to me that one also obtains another proof of Tukia’s conjecture (proved by Casson-Jungreis and Gabai) since you can cubulate a hyperbolic group with boundary a circle by cyclic subgroups.

That’s very interesting. Is it possible to say a bit more about how the hierarchy helps one to extend the action over the ball? It seems germane to point out that Tao Li gave examples of 3-manifolds that don’t admit 3-dimensional cubulations (as I think I learned in a talk that you gave, Ian!).

@ Henry (I’m not sure why I can’t reply to your comment, so I’ll reply to mine!): The idea would be to pass to a cover in which all of the surfaces embed and are acylindrical (i.e. pass to a subgroup in which all of the surface subgroups are malnormal). Then you create a hierarchy inductively out of the embedded surfaces, cutting along each one inductively. Two embedded surfaces intersect in a subsurface, so one uses pieces of the complements of these subsurfaces to create the hierarchy. So it doesn’t actually give a 3-dimensional cubulation, even of a finite-sheeted cover. In any case, I should attribute this approach to Markovich. When I had thought of this originally, I was contemplating using some sort of abstract hierarchy coming from a (high-dim) cubulation, and carrying out Thurston’s geometrization of Haken manifolds using the skinning map etc. But Vlad pointed out to me that one need only show that a finite-index subgroup extends as a group action over the ball.

Vlad has written a nice paper explaining how if one could construct “enough” quasi-convex surface subgroups of a hyperbolic group with 2-sphere ideal boundary, then one could prove it is virtually a Kleinian group. Enough means for any pair of points on the sphere, there is a circle limit set of a quasi-convex surface subgroup which separates the two points. http://front.math.ucdavis.edu/1205.5747

Of course, proving the existence of quasi-convex surface groups still is quite non-trivial. In Kahn-Markovic’s construction, they make use of some delicate geometric estimates to get the surfaces to be nearly geodesic, and they make use of the strong mixing of the frame flow in a hyperbolic manifold, which is a result due to Moore and makes use of representation theory of SL(2,C). So it’s not yet clear whether this will be a viable approach to the Cannon conjecture. But hopefully it will encourage people to think more about Gromov’s conjecture about surface subgroups of hyperbolic groups. Note that a result of Kapovich-Kleiner would imply that hyperbolic groups with Sierpinski carpet ideal boundary would be Kleinian groups if Cannon’s conjecture is true. http://www.sciencedirect.com/science/article/pii/S0012959300010491

Thanks for pointing this out, Ian. Incidentally, as far as I can tell, if Gromov conjectured anything, it was that there are one-ended word-hyperbolic groups without surface subgroups (see p. 144 of ‘Hyperbolic Groups’, Essays in group theory…). Of course, it would be interesting to know if he had conjectured something else somewhere else.

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